E Module - Taylor Polynomials

Question 1

What factors effect the error in a Taylor Polynomial and why?

Answer 1

The degree of the taylor polynomial n - As n increases the better the approximating is

Where is polynomial is centered, a - the closer the center, the closer we are to the true value at x

The behaviour of the function - we are only looking at the centered point a. Therefore, there could be a huge difference in behaviour from the point a to x.

Question 2

What is an infinite sum? When does it converge and when does it diverge?

Answer 2

An “infinite sum” is called a “series” in mathematics. In general, a series can either converge or diverge. If it converges, we’d love to know what it converges to. For example:

If the limit does exists and is finite, we say “The series converges”.
If the limit doesn’t exist or is infinite, we say ‘‘The series diverges”. (refer to E2 PCE notes for better format)

Question 3

Taylor’s Theorem

Answer 3

Taylor’s Theorem: Choose a centre x = a . If f(x) has (n+1) continuous derivatives on an interval around x=a, then for any x in that interval, we can write:

f(x) = pn(x) - Rn(x)

where pn(x) is the n-th Taylor Polynomial of f(x) centred at x=a .

and Rn(x) = (f(n+1)(c)/(n+1)! )(x-a)^n+1 for some c between x and a.

Note: The c in the remainder formula depends on both a and x. If you change either x or a , then c will likely change as well.

Question 4

What is the point of Taylor’s Theorem

Answer 4

To figure out how many taylor for which nth taylor polynomial should be taken for a certian error or to figure out which error is given for a certain nth taylor polynomial

Question 5

How do we find c?

Answer 5

c is between a and x. We don’t know the exact c, but can estimate it. This will give us an upper bound on the error approximation.

use the properties of the function to find an upper bound

ex1 - For example, imagine that we want to approximate e^0.9 using the Taylor polynomial for e^x of order 3 centered at 0. We get the approximation

R3(0.9) = (e^c/4!)x^4

for some c between 0 and 0.9. We know e^0

Question 6

What are Taylor Polynomials? Why do we use them?

Answer 6

Taylor polynomials are used to approximate really complicated functions with polynomials since polynomials are a lot easier to work with. In MAT186 we approximated tangent lines. With Taylor Polynomials, we can approximate the function with a parabola or cubic or quartic…

Question 7

What are the steps to creating a Taylor Polynomial? What is the general formula?

Answer 7

general formula:

pn(x) = (f^(n)(c)/n!))(x-a)^n

1 - commit to a point x = a. This is where ur polynomial will be “centered” aka the point of perfection because this is where the approximation will be perfect yasss

2 - plug into formuler and solve

The higher the degree of our taylor polynomial, the better the approximation